De-biasing the Lasso: Optimal Sample Size for Gaussian Designs (1508.02757v3)

Abstract: Performing statistical inference in high-dimension is an outstanding challenge. A major source of difficulty is the absence of precise information on the distribution of high-dimensional estimators. Here, we consider linear regression in the high-dimensional regime $p\gg n$. In this context, we would like to perform inference on a high-dimensional parameters vector $\theta^*\in{\mathbb R}^p$. Important progress has been achieved in computing confidence intervals for single coordinates $\theta^*_i$. A key role in these new methods is played by a certain debiased estimator $\hat{\theta}^{\rm d}$ that is constructed from the Lasso. Earlier work establishes that, under suitable assumptions on the design matrix, the coordinates of $\hat{\theta}^{\rm d}$ are asymptotically Gaussian provided $\theta^*$ is $s_0$-sparse with $s_0 = o(\sqrt{n}/\log p )$. The condition $s_0 = o(\sqrt{n}/ \log p )$ is stronger than the one for consistent estimation, namely $s_0 = o(n/ \log p)$. We study Gaussian designs with known or unknown population covariance. When the covariance is known, we prove that the debiased estimator is asymptotically Gaussian under the nearly optimal condition $s_0 = o(n/ (\log p)^2)$. Note that earlier work was limited to $s_0 = o(\sqrt{n}/\log p)$ even for perfectly known covariance. The same conclusion holds if the population covariance is unknown but can be estimated sufficiently well, e.g. under the same sparsity conditions on the inverse covariance as assumed by earlier work. For intermediate regimes, we describe the trade-off between sparsity in the coefficients and in the inverse covariance of the design. We further discuss several applications of our results to high-dimensional inference. In particular, we propose a new estimator that is minimax optimal up to a factor $1+o_n(1)$ for i.i.d. Gaussian designs.

• The paper introduces an optimal sparsity condition (s₀ = o(n/(log p)²)) that enhances the inference capabilities of debiased Lasso estimators in high-dimensional settings.
• It demonstrates that both known and estimated covariance scenarios satisfy the asymptotic Gaussian property under the new conditions.
• The authors validate their approach with numerical simulations, confirming that refined sample size requirements lead to more reliable statistical inference.

Debiasing the Lasso: Optimal Sample Size for Gaussian Designs

In the paper "Debiasing the Lasso: Optimal Sample Size for Gaussian Designs," the authors investigate the statistical inference challenges associated with high-dimensional models, specifically focusing on the Lasso estimator within linear regression contexts where the parameter dimension $p$ greatly exceeds the sample size $n$. This scenario is critical for many applications in machine learning and data science, where high-dimensional data is ubiquitous yet traditional inferential methods fall short due to overfitting concerns.

Main Contributions

The paper addresses the limitations of traditional Lasso estimators concerning inference tasks, such as constructing confidence intervals and calculating valid p-values for individual coefficients in high-dimensional settings. Building on previous work regarding debiased or de-sparsified estimators, the authors propose improved conditions under which the debiased Lasso achieves asymptotic Gaussian distributions, thereby facilitating reliable inference.

Key Results

1. Optimal Conditions for Debiased Estimator:
   • The manuscript details the conditions under which debiased estimators are approximately Gaussian. It introduces a novel condition for sparsity, $s_0 = o(n/(\log p)^2)$, which improves upon the previously established $s_0 = o(\sqrt{n}/\log p)$, thus allowing for greater sparsity while achieving similar inferential power.
2. Known and Unknown Covariance Scenarios:
   • When the population covariance matrix is known, the debiased estimator guarantees asymptotic Gaussianity under the proposed optimal sparsity condition. Furthermore, when the covariance matrix is unknown but can be sufficiently estimated, comparable results are achieved, leveraging sparseness assumptions for the inverse covariance matrix.
3. Numerical Validation and Comparison:
   • The authors validate their theoretical findings through simulations, demonstrating practical improvements of the proposed conditions over existing methods. They further illustrate the relationship between sample size requirements and sparsity, supporting their theoretical contributions.

The paper enhances the understanding of debiased Lasso estimators by establishing sample size requirements more rigorously and demonstrates through theoretical analysis and numerical experiments that these new conditions provide statistical guarantees that were previously unavailable.

Implications

The implications of these findings are significant for both theoretical and applied statistics within high-dimensional frameworks. Practically, the ability to construct reliable confidence intervals and perform hypothesis testing in settings where $p \gg n$ is invaluable across diverse fields, including genomics, finance, and advanced machine learning applications. Theoretically, the integration of refined sparsity conditions for precise inference marks a substantial contribution to statistical methodologies in high-dimensional data analysis.

Future Directions

Building upon these insights, future research may explore the extension of debiasing techniques to broader classes of statistical models beyond linear regression. There is potential to further investigate non-Gaussian designs and complex dependencies among predictor variables, thereby broadening the scope of application and tackling diverse real-world problems. Moreover, the trade-offs between sample size, sparsity level, and computational feasibility present fruitful areas for exploration, particularly in the domain of scalable machine learning solutions.

In conclusion, the paper presents significant advancements in the field of high-dimensional statistical inference, offering enhanced methodologies for debiasing the Lasso with implications that extend across multiple disciplines reliant on data-driven insights.